Lecture 9: Synthetic Control Methods

class: center, middle, inverse, title-slide

.title[
# Lecture 9: Synthetic Control Methods
]
.author[
### James Sears* <br> AFRE 891 SS 24 <br> Michigan State University
]
.date[
### .small[<br> *Parts of these slides are adapted from <a href="https://github.com/edrubin/EC607S23">“Econometrics III”</a> by Ed Rubin.]
]

---

# Table of Contents

1. [Prologue](#prologue)

3. [Matching](#match)

4. [Canonical Synthetic Control](#canon)

5. [Synthetic Difference-in-Difference](#synthdid)

6. [Partially Pooled SCM](#pool)
  
---
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name: prologue

# Prologue

---
# Prologue

This lecture is focusing on .hi-medgrn[Synthetic Control Methods], which will let us solve several of the issues that can affect methods we discussed last lecture.

.pull-left[
.center.hi-purple[Part 1]
  * The Fundamental Problem of Causal Inference
  * Matching
  * Canonical Synthetic Control
]

.pull-right[
.center.hi-pink[Part 2]
  * Synthetic Diff-in-Diff
    * Uniform Adoption
    * Staggered Adoption
  * Partially Pooled Synthetic Control
]

---
# Prologue

Packages we'll use today:

```r
# If not installed, add in packages from GitHub not on CRAN
if (!require("augsynth")) remotes::install_github("ebenmichael/augsynth")
```

```r
if (!require("pacman")) install.packages("pacman")
pacman::p_load(augsynth, fixest, synthdid, tidysynth, tidyverse)
```

As well, let's load the event study data from [Sears et al. (2023)](https://doi.org/10.1086/721705) we finished last lecture with:

```r
sah <- readRDS("data/sah_es.rds")
```

---
# Causal Inference

Let's chat about the .hi-blue[fundamental problem of causal inference] for a moment.

Consider unit `$i$`'s .hi-medgrn[potential outcomes:]
  
  * `$\text{Y}_{1i}$`: the outcome for unit `$i$` under the treatment
    * Treatment assignment `$D_i = 1$`
  * `$\text{Y}_{0i}$`: the outcome for unit `$i$` absent the treatment
    * Treatment assignment `$D_i = 0$`

1. We want/need to know `$\tau_i = \text{Y}_{1i} - \text{Y}_{0i}$`.
2. We cannot simultaneously observe *both* `$\text{Y}_{1i}$` *and* `$\text{Y}_{0i}$`.

Most (all?) empirical strategies boil to estimating `$\text{Y}_{0i}$` for treated individuals — the .hi-green[unobservable counterfactual] for the treatment group.

---
# Causal Inference

Last lecture gave an overview of regression methods that make different assumptions about that .hi-green[unobservable counterfactual].

.hi-blue[1\. RCT + Random Assignment]
  * The average in the control group is what the average in the treatment group would have been absent the treatment
  * Regress outcome on treatment dummy and you're good to go
    * Maybe add some control variables to improve precision of estimator
--

<br>

.hi-blue[2\. Difference-in-Differences + Event Study]

* The .hi-medgrn[change over time] in the control group is what the change in the treatment group would have been absent the treatment

---
# Causal Inference

All of these estimates are identified under a variation of the .hi-purple[Conditional Independence Assumption (CIA)].super[.pink[†]]

$$
`\begin{align}
  \def\ci{\perp\mkern-10mu\perp}
  \left\{ \color{#6A5ACD}{\text{Y}_{0i}},\,\color{#6A5ACD}{\text{Y}_{1i}} \right\} \ci  \color{#e64173}{\text{D}_{i}} ~|~ \text{X}_{i}
\end{align}`
$$
--

Conditional on `$\text{X}_{i}$`.super[1], potential outcomes `$\left( \color{#6A5ACD}{\text{Y}_{0i}},\, \color{#6A5ACD}{\text{Y}_{1i}} \right)$` are independent of treatment status `$\left( \color{#e64173}{\text{D}_{i}} \right)$`.

.footnote[.pink[†] AKA "selection on observables". <br> 1. Or *it* if we're in a panel setting]

---
# Causal Inference

But there are times when .hi-pink[CIA fails].super[2].

.footnote[2\. See the [Bay of Pigs Invasion](https://www.jfklibrary.org/learn/about-jfk/jfk-in-history/the-bay-of-pigs)]

In the case of Diff-in-Diff and Event Study, this is often due to a failure of .hi-purple[parallel trends].

For example, recall the event study for mobility responses to stay-at-home mandates from last lecture:

---
# Causal Inference

In this case, the states that never adopted stay-at-home mandates might not be *valid counterfactuals* for the states that did adopt.

However, there might be a way to *construct* a valid counterfactual from the set of control units...

---
# Causal Inference

... but before we get into that, let's chat briefly about .hi-purple[matching estimators].

---
class: inverse, middle
name: match

# Matching

---

# Matching

.hi-purple[Matching Estimators] provide an alternate way of coming up with the unobservable counterfactual for the treatment group.

.hi-medgrn[The gist:]

* Match untreated observations to treated observations using `$\text{X}_{i}$`
    * _i.e._ calculate a `$\widehat{\text{Y}_{0i}}$` for each `$\text{Y}_{1i}$`, based upon "matched" untreated individuals with (nearly) identical values of `$X_i$`
  * If CIA holds, then we can just calculate a bunch of treatment effects conditional on `$\text{X}_{i}$`
    * _i.e._ 
    
$$
`\begin{align}
  \tau(x) = \mathop{E}\left[ \text{Y}_{1i} - \text{Y}_{0i} \mid \text{X}_{i} = x \right]
\end{align}`
$$

---

# Matching

.hi-blue[More formally:]

We want to construct a counterfactual for each individual with `$\text{D}_{i}=1$`.

.note[CIA:] The counterfactual for `$i$` should only use individuals that match `$\text{X}_{i}$`.

Let there be `$N_T$` treated individuals and `$N_C$` control individuals. We want
- `$N_T$` sets of weights
- with `$N_C$` weights in each set
--
: `$w_i(j)\, \left( i = 1,\,\ldots,\, N_T;\, j=1,\,\ldots,\, N_C \right)$`

Assume `$\sum_j w_i(j) = 1$`. Our estimate for the counterfactual of treated `$i$` is
$$
`\begin{align}
  \widehat{\text{Y}_{0i}} = \sum_{j\in \left( D=0 \right)} w_i(j) \text{Y}_{j}
\end{align}`
$$

---
# Weight for it

So all we need is those weights and we're done.

.hi-medgrn[Q:] Where does one find these handy weights?

.hi-blue[A:] You've got options, but you need to choose carefully/responsibly.

*E.g.* if `$w_i(j) = \frac{1}{N_C}$` for all `$(i,j)$`, then we're back to a difference in means.
<br> This weighting doesn't abide by our conditional independence assumption.

.hi-green[The plan:] choose weights `$w_i(j)$` that indicate .hi-slate[*how close*] `$\text{X}_{j}$` is to `$\text{X}_{i}$`.

---
# Weight for it

Some common choice of weights:

* .hi-medgrn[Nearest neighbor:]
$$
`\begin{align}
  \text{d}_{i,j} = \left( \text{X}_{i} - \text{X}_{j} \right)'\left(\text{X}_{i} - \text{X}_{j}\right)
\end{align}`
$$
  * .hi-blue[Kernel Matching] for .hi-slate[bandwidth] `$h$` and .hi-slate[kernel function] `$K(\cdot)$`:
$$
`\begin{align}
  w_i(j) = \dfrac{K\!\!\left( \dfrac{\text{X}_{j} - \text{X}_{i}}{h} \right)}{\sum\limits_{j\in(D=0)} K\!\!\left(\dfrac{\text{X}_{j} - \text{X}_{i}}{h} \right)}
\end{align}`
$$

---
# Kernels

For example, the *Epanechnikov kernel* is defined as
$$
`\begin{align}
  K(z) = \dfrac{3}{4} \left( 1 - z^2 \right) \times \mathbb{I}\!\left( |z| < 1 \right)
\end{align}`
$$

---
# Kernels

And the *triangular kernel* can be expressed as  `$K(z) = \left( 1 - |z| \right) \times \mathbb{I}\!\left( |z| < 1 \right)$`

<img src="data:image/png;base64,#09-Synthetic_files/figure-html/triangle-1.png" width="80%" style="display: block; margin: auto;" />
---
# Kernels

And the *uniform kernel* with `$K(z) = \frac{1}{2} \times \mathbb{I}\!\left( |z| < 1 \right)$`

<img src="data:image/png;base64,#09-Synthetic_files/figure-html/uniform-1.png" width="80%" style="display: block; margin: auto;" />
---
# Kernels

Or the *Gaussian kernel* `$K(z) = \left( 2\pi \right)^{-1/2}  \exp\left(-z^2/2 \right)$`

<img src="data:image/png;base64,#09-Synthetic_files/figure-html/gaussian-1.png" width="80%" style="display: block; margin: auto;" />
---
# Aside on Kernels

Kernel functions are good for more than just matching.

You will most commonly see/use them smoothing out densities—providing a smooth, moving-window average.

_E.g._ .mono[R]'s (`ggplot2`'s) smooth, density-plotting function `geom_density()`.

`geom_density()` defaults to `kernel = "gaussian"`, but you can specify many other kernel functions (including `"epanechnikov"`).

You can also change the `bandwidth` argument. The default is a bandwidth-choosing function called `bw.nrd0()`.

---
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name: canon

# Canonical Synthetic Control